Consider a functional of the form

\[T(F)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} h\left(x_{1}, x_{2}ight) d F\left(x_{1}ight) d F\left(X_{2}ight)\]

where \(F \in \mathcal{F}\), a collection of distribution functions. Find an expression for the \(k^{\text {th }}\) Gâteaux differential of \(T(F)\) without using the assumption that \(h\) is a symmetric function of its arguments. Compare this result to that of Theorem 9.1.

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Theorem 9.1. Consider a functional of the form T(F) - L-L = r h(x1,...,xr) II dF (x;), i=1 where F = F, and F is a collection of distribution functions. Then if k r, k |AT(F; G F) = |||(r=i+1)]; L L .. h (x1,..., 2 X r-k dF (ya d[G(xi) F(xi)] - F(x;)]}. Tk, Y1,..., Yr-k) II If kr then AT(F; G-F) = 0. i=1